Question: What is the period of the function $g(x)=-9\cos\left(-\dfrac{\pi}{2} x-6\right)+8$ ? Give an exact value. units
Period in sinusoids of the form $y=a\cos(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\cos( bx + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $g(x) = -9\cos\left({-\dfrac{\pi}{2}}x-6\right)+8$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{\left|{-\dfrac{\pi}{2}}\right |} \\\\\\\\\\ &=\dfrac{2\pi}{\dfrac{\pi}{2}}\\\\\\\\ &= 2\pi\cdot \dfrac{2}{\pi} \\\\ &=4 \end{aligned}$ The answer The period of $g(x) = -9\cos\left({-\dfrac{\pi}{2}}x-6\right)+8$ is $4$ units.